PLEASE HELP WITH ALL WILL THANK AND MARK BRAINLIEST
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Answer:
E(A)= E[E(A|B)]= 4*0.6 +5*0.4 =4.4
Var(A)= E[Var(A|B)] +Var[E(X|Y)]]=4.4+19.6=24
Step-by-step explanation:
Previous concepts
The Poisson process is useful when we want to analyze the probability of ocurrence of an event in a time specified. The probability distribution for a random variable X following the Poisson distribution is given by:
For this distribution the expected value is the same parameter
Other equations useful:
Let X and Y random variables
E(X) =E[E(X|Y)] (conditional expectation)
Var(X)=E[Var(X|Y)]+Var[E(X|Y)] (Total variance)
Solution to the problem
Let A the random variable that represent the number of winter storms next year
B a binary variable, B=1 if the next year is a good year and B=0 in the other case. Then we have this:
E(A|B=1) = 4 and E(A|B=0)=5
We can use the propoerties of conditional expectation like this:
E(A)= E[E(A|B)]= E(A|B=1)P(B=1) +E(A|B=0)P(B=0)
E(A)= E[E(A|B)]= 4*0.6 +5*0.4 =4.4
And we can use also the properties for conditional variance we have the following values:
Var(A|B=1)=4 Var(A|B=0)=5, by the propertis of the Poisson distribution
And then the conditional variance is givne by:
And if we replace we got:
And we have also that the expected value for the conditional variance is given by:
E[Var(A|B)]= 4*0.6 +5*0.4 =4.4
And then finally the variance for the random variable A is given by:
Var(A)= E[Var(A|B)] +Var[E(X|Y)]]=4.4+19.6=24